Pattern Formation in Wigner-like Equations via Multiresolution
Antonina N. Fedorova, Michael G. Zeitlin
备注：
15 pages, 6 figures, ws-procs9x6.cls, Presented at Joint 28th ICFA Advanced Beam Dynamics & Advanced & Novel Accelerators Workshop on Quantum Aspects of Beam Physics and Other Critical Issues of Beams in Physics and Astrophysics, January 7-11, 2003, Hiroshima University, Higashi-Hiroshima, Japan
We present the application of the variational-wavelet analysis to the quasiclassical calculations of the solutions of Wigner/von Neumann/Moyal and related equations corresponding to the nonlinear (polynomial) dynamical problems. (Naive) deformation quantization, the multiresolution representations and the variational approach are the key points. We construct the solutions via the multiscale expansions in the generalized coherent states or high-localized nonlinear eigenmodes in the base of the compactly supported wavelets and the wavelet packets. We demonstrate the appearance of (stable) localized patterns (waveletons) and consider entanglement and decoherence as possible applications.
We demonstrate the creation of nontrivial (meta) stable states (patterns), localized, chaotic, entangled or decoherent, from the basic localized modes in various collective models arising from the quantum hierarchy described by Wigner-like equations. The numerical simulation demonstrates the formation of various (meta) stable patterns or orbits generated by internal hidden symmetry from generic high-localized fundamental modes. In addition, we can control the type of behavior on the pure algebraic level by means of properly reduced algebraic systems (generalized dispersion relations).
Many practical sampling patterns for function approximation on the rotation group utilizes regular samples on the parameter axes. In this paper, we relate the mutual coherence analysis for sensing matrices that correspond to a class of regular patterns to angular momentum analysis in quantum mechanics and provide simple lower bounds for it. The products of Wigner d-functions, which appear in coherence analysis, arise in angular momentum analysis in quantum mechanics. We first represent the product as a linear combination of a single Wigner d-function and angular momentum coefficients, otherwis…
Phase-space features of the Wigner flow for generic one-dimensional systems with a Hamiltonian, $H^{W}(q,\,p)$, constrained by the $\partial ^2 H^{W} / \partial q \partial p = 0$ condition are analytically obtained in terms of Wigner functions and Wigner currents. Liouvillian and stationary profiles are identified for thermodynamic (TD) and Gaussian quantum ensembles to account for exact corrections due to quantum modifications over a classical phase-space pattern. General results are then specialized to the Harper Hamiltonian system which, besides working as a feasible test platform for the f…
We study the formation of lines in phase space in Wigner's distribution $W.$ Whereas lines in phase space do not form in classical systems, unless special initial states are chosen, we find, for large classes of systems and initial states of quantum systems that $W$ tends to form straight line patterns crisscrossing phase space. These arise from the states' coherences. Some of those lines have astonishing extent, reaching across the entire state. We show that the formation of such straight line patterns is due to the formation of 'randomized grid states'. We establish their stability to pertur…
Using Dirac equation together with the Wigner distribution function,the trembling motion,known as Zitterbewegung effect,of moving electrons in quasi-one-dimensional relativistic quantum plasma is theoretically investigated.The relativistic Wigner matrix is used to calculate the mean values of the position and velocity operators for a Dirac gas of electrons.It is found that the oscillatory behavior of measurable quantities could be associated with the Zitterbewegung effect which manifests itself as a damping interference pattern stemming from mixing the positive and negative dispersion modes of…
Exploiting the correspondence between Wigner distribution function and a frequency-resolved optical gating (FROG) measurement, we experimentally demonstrate existence of the chessboard-like interference patterns with a time-bandwidth product smaller than that of a transform-limited pulse in the phase space representation of compass states. Using superpositions of four electric pulses as realization of compass states, we have shown via direct measurements that displacements leading to orthogonal states can be smaller than limits set by uncertainty relations. In the experiment we observe an exac…
